# What is the formula for GaussianBeam?

| Date       | Category    |
|------------|-------------|
| 2025-09-05 17:29:25 | Sources |


The GaussianBeam source has the following scalar field amplitude, in cylindrical coordinates:
 
<center>
    $u(r,z)=\frac{w_0}{w(z)}e^{-\frac{r^2}{w(z)^2}}e^{i(zk_0 + \frac{r^2k_0}{2R(z)} - \psi_g)}$
</center><br>
where:
<ul>
    <li>$z$ is the propagation direction</li>
    <li>$k_0=\frac{2\pi nf}{c}$ are the wavenumbers of the frequencies $f$ where the beam is sampled</li>
    <li>$w_0$ is the beam waist</li>
    <li>$w(z)=w_0\sqrt{1 + \frac{(z + z_0)^2}{z_r}}$, where $z_0$ is the waist distance and $z_r=\frac{1}{2}w_0^2k_0$ is the Rayleigh range</li>
    <li>$R(z)=z(1 + (\frac{z_r}{z})^2)$ is the radius of curvature of the wavefront at $z$</li>
    <li>$\psi_g=\arctan(\frac{z+z_0}{z_r})-\arctan(\frac{z_0}{z_r})$ is the Gouy phase</li>
</ul>

See the code [here](https://docs.flexcompute.com/projects/tidy3d/en/latest/_modules/tidy3d/components/beam.html#GaussianBeamProfile).